Dispersion compensation

1. Dispersion

2. Why does dispersion matter ?
• Understanding the effects of dispersion in optical
fibers is quite essential in optical communications
in order to minimize pulse spreading.
• Pulse compression due to negative dispersion can
be used to shorten pulse duration in chirped pulse
lasers

3. Signal Dispersion
• Inter modal dispersion
– Only in Multimode fibers.
– Each mode having different value of group velocity
at single frequency
• Intramodal dispersion or Chromatic dispersion
– Within a single mode.
– Group velocity dispersion

5. Dispersion in Optical Fibers
• Dispersion: Any phenomenon in which the velocity of propagation of any
electromagnetic wave is wavelength dependent.
• In communication, dispersion is used to describe any process by which any
electromagnetic signal propagating in a physical medium is degraded
because the various wave characteristics (i.e., frequencies) of the signal
have different propagation velocities within the physical medium.
• There are 3 dispersion types in the optical fibers, in general:
1- Material Dispersion
2- Waveguide Dispersion
3- Polarization-Mode Dispersion
Material & waveguide dispersions are main causes of Intramodal
Dispersion.

6. Pulse broadening and attenuation

7. Group Velocity
• Wave Velocities:
• 1- Plane wave velocity: For a plane wave propagating along z-axis in an
unbounded homogeneous region of refractive index , which is
represented by , the velocity of constant phase plane is:
• 2- Modal wave phase velocity: For a modal wave propagating along z-axis
represented by , the velocity of constant phase plane is:
3- For transmission system operation the most important & useful type of
velocity is the group velocity, . This is the actual velocity which the
signal information & energy is traveling down the fiber. It is always less
than the speed of light in the medium. The observable delay experiences by
the optical signal waveform & energy, when traveling a length of l along
the fiber is commonly referred to as group delay.
1
n
)
ω
exp( 1z
jk
t
j 
1
1 n
c
k
v 


)
ω
exp( z
j
t
j 


ω

p
v
[3-4]
[3-5]
g
V

8. Group Velocity & Group Delay
• The group velocity is given by:
• The group delay is given by:
• It is important to note that all above quantities depend both on frequency
& the propagation mode. In order to see the effect of these parameters on
group velocity and delay, the following analysis would be helpful.

d
d
Vg
ω
 [3-6]
ω
d
d
l
V
l
g
g

 
 [3-7]

9. Input/Output signals in Fiber Transmission System
• The optical signal (complex) waveform at the input of fiber of length l is
f(t). The propagation constant of a particular modal wave carrying the
signal is . Let us find the output signal waveform g(t).
)
ω
(

z-=0 Z=l













c
c
d
e
f
t
f t
j
)
(
~
)
( [3-8]
















c
c
d
e
f
t
g l
j
t
j )
(
)
(
~
)
( [3-9]
bandwidth.
signal
optical
the
is



10. ...
)
(
2
1
)
(
)
(
)
(
If
2
2
2










c
c
c
c
c
c
d
d
d
d


















)
(
)
(
)
(
~
)
(
~
)
(
~
)
(
)
(
)
(
)
(
2
/
2
/
)
(
)]
(
)
(
[
2
/
2
/
)
(
2
/
2
/
g
l
j
l
j
d
d
l
t
j
l
j
l
d
d
j
t
j
l
j
t
j
t
f
e
d
d
l
t
f
e
d
e
f
e
d
e
f
d
e
f
t
g
c
c
c
c
c
c
c
c
c
c
c
c
c
c















































































g
g
V
l
d
d
l
c







[3-10]
[3-11]
[3-14]

11. Intramodal Dispersion
• As we have seen from Input/output signal relationship in optical fiber, the
output is proportional to the delayed version of the input signal, and the
delay is inversely proportional to the group velocity of the wave. Since the
propagation constant, , is frequency dependent over band width
sitting at the center frequency , at each frequency, we have one
propagation constant resulting in a specific delay time. As the output signal
is collectively represented by group velocity & group delay this
phenomenon is called intramodal dispersion or Group Velocity
Dispersion (GVD). This phenomenon arises due to a finite bandwidth
of the optical source, dependency of refractive index on the
wavelength and the modal dependency of the group velocity.
• In the case of optical pulse propagation down the fiber, GVD causes pulse
broadening, leading to Inter Symbol Interference (ISI).
)
ω
(
 ω

c
ω

12. Dispersion & ISI
Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000
A measure of information
capacity of an optical fiber for
digital transmission is usually
specified by the bandwidth
distance product
in GHz.km.
For multi-mode step index fiber
this quantity is about 20
MHz.km, for graded index fiber
is about 2.5 GHz.km & for single
mode fibers are higher than 10
GHz.km.
L
BW 

13. How to characterize dispersion?
• Group delay per unit length can be defined as:
• If the spectral width of the optical source is not too wide, then the delay
difference per unit wavelength along the propagation path is approximately
For spectral components which are apart, symmetrical around center
wavelength, the total delay difference over a distance L is:







d
d
c
dk
d
c
d
d
L
g
2
1
ω
2



 [3-15]


d
d g





















































2
2
2
2
2
2
2
d
d
L
V
L
d
d
d
d
d
d
d
d
c
L
d
d
g
g
[3-16]

14. • is called GVD parameter, and shows how much a light pulse
broadens as it travels along an optical fiber. The more common parameter
is called Dispersion, and can be defined as the delay difference per unit
length per unit wavelength as follows:
• In the case of optical pulse, if the spectral width of the optical source is
characterized by its rms value of the Gaussian pulse , the pulse
spreading over the length of L, can be well approximated by:
• D has a typical unit of [ps/(nm.km)].
2
2
2



d
d

2
2
2
1
1





 c
V
d
d
d
d
L
D
g
g











 [3-17]


g


 



 DL
d
d g
g 
 [3-18]

15. 
t
Spread, ² 
t
0

Spectrum, ² 
1
2
o
Intensity Intensity Intensity
Cladding
Core
Emitter
Very short
light pulse
vg
(2
)
vg
(1
)
Input
Output
All excitation sources are inherently non-monochromatic and emit within a
spectrum, ² , of wavelengths. Waves in the guide with different free space
wavelengths travel at different group velocities due to the wavelength dependence
of n1. The waves arrive at the end of the fiber at different times and hence result in
a broadened output pulse.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Material Dispersion

16. Material Dispersion
• The refractive index of the material varies as a function of wavelength,
• Material-induced dispersion for a plane wave propagation in homogeneous
medium of refractive index n:
• The pulse spread due to material dispersion is therefore:
)
(
n

































d
dn
n
c
L
n
d
d
L
c
d
d
L
c
d
d
L
mat )
(
2
2
2
ω
2
2
[3-19]
)
(
2
2








 

 mat
mat
g D
L
d
n
d
c
L
d
d


 [3-20]
)
(
mat
D is material dispersion

17. Material Dispersion Diagrams
Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000

18. Waveguide Dispersion
• Waveguide dispersion is due to the dependency of the group velocity of the
fundamental mode as well as other modes on the V number, (see Fig 2-18
of the textbook). In order to calculate waveguide dispersion, we consider
that n is not dependent on wavelength. Defining the normalized
propagation constant b as:
• solving for propagation constant:
• Using V number:
2
1
2
2
2
2
1
2
2
2
2
/
/
n
n
n
k
n
n
n
k
b







 [3-21]
)
1
(
2 

 b
k
n
 [3-22]



 2
)
( 2
2
/
1
2
2
2
1 kan
n
n
ka
V [3-23]

19. Waveguide Dispersion
• Delay time due to waveguide dispersion can then be expressed as:









dV
Vb
d
n
n
c
L
wg
)
(
2
2
 [3-24]
Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000

20. Waveguide dispersion in single mode fibers
• For single mode fibers, waveguide dispersion is in the same order of
material dispersion. The pulse spread can be well approximated as:
2
2
2 )
(
)
(
dV
Vb
d
V
c
L
n
D
L
d
d
wg
wg
wg







 





 [3-25]
Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000
)
(
wg
D

21. Polarization Mode dispersion
Core
z
n1x
// x
n1y
// y
Ey
Ex
Ex
Ey
E
 = Pulse spread
Input light pulse
Output light pulse
t
t

Intensity
Suppose that the core refractive index has different values along two orthogonal
directions corresponding to electric field oscillation direction (polarizations). We can
take x and y axes along these directions. An input light will travel along the fiber with Ex
and Ey polarizations having different group velocities and hence arrive at the output at
different times
© 1999 S.O. Kasap,Optoelectronics (Prentice Hall)

22. Polarization Mode dispersion
• The effects of fiber-birefringence on the polarization states of an optical are
another source of pulse broadening. Polarization mode dispersion (PMD)
is due to slightly different velocity for each polarization mode because of
the lack of perfectly symmetric & anisotropicity of the fiber. If the group
velocities of two orthogonal polarization modes are then the
differential time delay between these two polarization over a
distance L is
• The rms value of the differential group delay can be approximated as:
gy
gx v
v and
pol


gy
gx
pol
v
L
v
L


 [3-26]
L
DPMD
pol 
 [3-27]

23. Chromatic & Total Dispersion
• Chromatic dispersion includes the material & waveguide dispersions.
• Total dispersion is the sum of chromatic , polarization dispersion and other
dispersion types and the total rms pulse spreading can be approximately
written as:





L
D
D
D
D
ch
ch
wg
mat
ch
)
(
)
(



[3-28]


 L
D
D
D
D
total
total
pol
ch
total



 ...
[3-29]

24. The magnitudes of material and waveguide dispersion as a
function of optical wavelength for a single-mode fused-silica-
core fiber.

25. Optimum single mode fiber & distortion/attenuation
characteristics
Fact 1) Minimum distortion at wavelength about 1300 nm for single mode
silica fiber.
Fact 2) Minimum attenuation is at 1550 nm for sinlge mode silica fiber.
Strategy: shifting the zero-dispersion to longer wavelength for minimum
attenuation and dispersion by Modifying waveguide dispersion by
changing from a simple step-index core profile to more complicated
profiles. There are four major categories to do that:
1- 1300 nm optimized single mode step-fibers: matched cladding (mode
diameter 9.6 micrometer) and depressed-cladding (mode diameter about 9
micrometer)
2- Dispersion shifted fibers.
3- Dispersion-flattened fibers.
4- Large-effective area (LEA) fibers (less nonlinearities for fiber optical
amplifier applications, effective cross section areas are typically greater
than 100 ).
2
m


26. Representative cross sections of index profi les for (a) 1310-nm-optimized,
(b) dispersion-shifted, (c) dispersion-fl attened, and (d) large-effective-core-area
fi bers

27. Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000
Typical waveguide dispersions and the common material dispersion for three
different single-mode fiber designs

28. Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000
Resultant total dispersions

29. Single mode Cut-off wavelength & Dispersion
• Fundamental mode is with V=2.405 and
• Dispersion:
• For non-dispersion-shifted fibers (1270 nm – 1340 nm)
• For dispersion shifted fibers (1500 nm- 1600 nm)
01
11 LP
or
HE 2
2
2
1
2
n
n
V
a
c 



[3-30]









L
D
D
D
d
d
D wg
mat
)
(
)
(
)
(
)
(



 [3-31]
[3-32]

30. Dispersion for non-dispersion-shifted fibers
(1270 nm – 1340 nm)
• is relative delay minimum at the zero-dispersion wavelength , and
is the value of the dispersion slope in .
2
2
0
0
0 )
(
8
)
(





 


S
0
 0
 0
S
.km)
ps/(nm2
0
)
( 0
0







d
dD
S
S
[3-33]
[3-34]







 4
0
0
)
(
1
4
)
(




S
D [3-35]

31. Dispersion for dispersion shifted fibers (1500
nm- 1600 nm)
2
0
0
0 )
(
2
)
( 



 


S
0
0 )
(
)
( S
D 

 

[3-36]
[3-37]

32. Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000
Example of dispersion
Performance curve for
Set of SM-fiber

33. Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000
Example of BW vs wavelength for various optical sources for
SM-fiber.

34. MFD
Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000

35. Bending Loss
Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000

36. Bending effects on loss vs MFD
Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000

37. Bend loss versus bend radius
Optical Fiber communications, 3rd ed.,G.Keiser,McGrawHill, 2000
07
.
0
;
10
56
.
3
m
60
;
m
6
.
3
2
2
3
3








n
n
n
b
a 


38. Kerr effect
I
n
n
n 2
0 
 Kerr nonlinearity in fiber, where I is the intensity of
Optical wave.
Temporal changes in a narrow optical pulse that is subjected to Kerr nonlinearity in
A dispersive medium with positive GVD.

39. First-order Soliton
Temporal changes in a medium with Kerr nonlinearity and negative GVD. Since dispersion tends to broaden the pulse, Kerr
Nonlinearity tends to squeeze the pulse, resulting in a formation of optical soliton.